Question: Simplify; express your answer in exponential form. Assume $q\neq 0, x\neq 0$. $\dfrac{{(qx^{-3})^{3}}}{{(q^{4}x^{-4})^{-1}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(qx^{-3})^{3} = (q)^{3}(x^{-3})^{3}}$ On the left, we have ${q}$ to the exponent ${3}$ . Now ${1 \times 3 = 3}$ , so ${(q)^{3} = q^{3}}$ Apply the ideas above to simplify the equation. $\dfrac{{(qx^{-3})^{3}}}{{(q^{4}x^{-4})^{-1}}} = \dfrac{{q^{3}x^{-9}}}{{q^{-4}x^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{3}x^{-9}}}{{q^{-4}x^{4}}} = \dfrac{{q^{3}}}{{q^{-4}}} \cdot \dfrac{{x^{-9}}}{{x^{4}}} = q^{{3} - {(-4)}} \cdot x^{{-9} - {4}} = q^{7}x^{-13}$